if you have VERY pure water then it would be a poor conductor and you could say that it would not conduct electricity. However even small amounts of salts (or other impurities) will increase electrical conductance.

As for the electrical appliance in the ocean, i'm not too sure. Maybe somebody else can answer this.

PURE water, in the purest sense, does not conduct electricity.
Water needs to have some type of electrolyte in order to allow an electrical current to be conducted through.
Distilled water should not conduct any significant amount of electrical current.

Electricity (movement of charged particles) flows between two sources of electrical potential.
If you dropped a toaster into the ocean, the electrical potential across the toaster is looking for the path of least resistance to the ground. The easiest way would be to head straight over to the opposite electrode in the toaster, short circuiting it. But there are also other path's the electricity can take to get to ground...it could take the long way and go through the entire ocean until it reaches the bottom or land, but this is a far more complicated path for the electricity to take and only a fraction of the total current would take this path. Most of the current should stay within the toaster, between the two electrodes I would think.

I have heard to argument made that even pure water conducts to a degree since in reality, "pure" water does not exist.
liquid water exists in a state of equilibrium between H+ and OH- ions in solution withitself,
H2O (l) <---> H+ (aq) + OH- (aq)

Also, CO2 from the air will dissolve in the water to form Carbonic acid which will weakly dissassociate into H+ and CO3- ions, providing for the conduction of an electric current.

I have never known anyone to fill up a bath tub with distilled water...most people use tap water. Tap water has certain ions dissolved in it, some are put there intentionally by the city (F- for example), and others get there unintentionally (Ca + 2 for example). These ions allow for the conduction of an electric current.

The ion concentration of tap water is not very high, it still has a significant electrical resistance, but never the less, when applying high voltages to the water, such as those you might find in an electrical appliance, a current will conduct.
V = RI

Current kills….and it does not take all that much current to kill you if it passes through the right part of your body.

That's at the very best, a poor argument. I hope you didn't read it in a book. If "pure" water doesn't exist, how can someone make a statement about its properties?

No, I dont think I read it in a book......but I do specifically remember hearing my PhD possessing Chemistry professor state it as fact at least once.
Just another reason I hate my school's chemistry department.

I have never seen any reluctance for the scientific community to speak of theoretical properties of things that are either very difficult or impossible to obtain. "Pure" water could well be considered in this category. Anyone who has spent any time around a commercial DI water system is aware of how difficult it is to maintain a resistivity of even 14Mohm -cm DI. Distilled water is NOT DeIonized so will have a relatively high conductivity on the scale of DI systems. Many commercial wet benches routinely use DI by the hundreds of galleons, so a bathtub full is not all that remarkable, as long as the bathtub is made of PTFE or some similar plastic.

The reason a bathtub full of soapy water or even tap water is an excellent conductor is because it is NOT pure, in fact it is a LONG way from being pure. I believe MrJeffery's prof, pure water is in fact non existent.